6) \( \begin{array}{l}y=5 x-7 \\ -3 x-2 y=-12\end{array} \)

To solve the given system of equations, we can start by rewriting the second equation in a more manageable form. The equation \(-3x - 2y = -12\) can be rearranged to express \(y\) in terms of \(x\). Dividing the entire equation by -2 gives us \(y = \frac{3}{2}x + 6\). Now, we have two equations: 1. \(y = 5x - 7\) 2. \(y = \frac{3}{2}x + 6\) Next, we can set the two equations equal to each other to find the value of \(x\): \[5x - 7 = \frac{3}{2}x + 6\] To clear the fractions, multiply every term by 2: \[10x - 14 = 3x + 12\] Now, rearranging this gives us: \[10x - 3x = 12 + 14\] \[7x = 26 \Rightarrow x = \frac{26}{7}\] Now plug \(x\) back into one of the original equations to get \(y\): Using \(y = 5x - 7\): \[y = 5\left(\frac{26}{7}\right) - 7\] \[y = \frac{130}{7} - \frac{49}{7} = \frac{81}{7}\] Thus, the solution to the system of equations is \(\left(\frac{26}{7}, \frac{81}{7}\right)\). This method not only provides the solution but reinforces the concept of substitution and equality of equations to find points of intersection, making it a valuable tool for various applications in science and engineering.